Modeling dependent time to visual acuity recovery using a bivariate copula Cox approach: a study of bilateral cataract surgery

Abstract

Survival analysis is a statistical framework for modeling time-to-event data. In the context of bilateral diseases such as cataracts, the two eyes may exhibit dependent event times, thereby reducing the suitability of univariate approaches for capturing this dependence. This study aims to model the dependence between the times to achieve a minimum visual acuity of 6/18 in the right and left eyes of patients following cataract surgery using a Cox proportional hazards model with a bivariate copula approach. The data consists of patients who underwent bilateral cataract surgery at an eye hospital in Surabaya, between January 1 and October 31, 2025. Parameter estimation was conducted using a two-stage procedure based on maximum pseudo-likelihood, with numerical optimization performed via the Berndt–Hall–Hall–Hausman (BHHH) algorithm. Four Archimedean copula families (Clayton, Frank, Gumbel, and Joe) were considered to model the dependence structure. Based on the Akaike Information Criterion (AIC), the Joe copula provided the best model fit, yielding an AIC value of 2975.158. The estimated dependence parameter was 3.934, corresponding to Kendall’s tau of 0.609, indicating a moderately strong positive dependence between the times to achieve the minimum visual acuity in both eyes. Additionally, age and the history of cardiovascular disease were found to have significant effects on the event times in both eyes.